The only thing more consistent than the benefits exercise has brought to my mental health, is the strange irrational hatred some people have for the idea that it can help.
Simply, it's not helpful. Fat people know they're fat because they have a caloric surplus. They've tried eating less. They're not stupid.
I think the defensiveness people exhibit is a byproduct of their cumulative reactions to a bunch of unsolicited advice. My own (unsolicited) advice would be: don't give people advice, unless they specifically ask for it. I can guarantee you that every depressed person on the planet has heard that exercise is good for your mental health. Instead, you can say something like, "I'm very sorry that you're having a hard time. I want you to know that you're important to me, and if there's ever anything I can do to help, let me know"